Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-2x-y &= 1 \\ 3x-6y &= 3\end{align*}$
Explanation: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $2$ $\begin{align*}-6x-3y &= 3\\ 6x-12y &= 6\end{align*}$ Add the top and bottom equations. $-15y = 9$ Divide both sides by $-15$ and reduce as necessary. $y = -\dfrac{3}{5}$ Substitute $-\dfrac{3}{5}$ for $y$ in the top equation. $-2x+ \dfrac{3}{5} = 1$ $-2x+\dfrac{3}{5} = 1$ $-2x = \dfrac{2}{5}$ $x = -\dfrac{1}{5}$ The solution is $\enspace x = -\dfrac{1}{5}, \enspace y = -\dfrac{3}{5}$.